Ibn-al-Haitam, al-Hasan Ibn-al-Hasan; Witelo; Risner, Friedrich, Opticae thesavrvs Alhazeni Arabis libri septem, nunc primùm editi. Eivsdem liber De Crepvscvlis & Nubium ascensionibus. Item Vitellonis Thuvringopoloni Libri X. Omnes instaurati, figuris illustrati & aucti, adiectis etiam in Alhazenum commentarijs, a Federico Risnero, 1572

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        <div xml:id="echoid-div386" type="section" level="0" n="0">
          <pb o="165" file="0171" n="171" rhead="OPTICAE LIBER V."/>
        </div>
        <div xml:id="echoid-div388" type="section" level="0" n="0">
          <head xml:id="echoid-head369" xml:space="preserve" style="it">66. Viſ{us} & uiſibile in diuerſis dimetris circuli (qui eſt commu nis ſectio ſuperficierum refle-
            <lb/>
          xionis & ſpeculi ſphærici caui) inter ſe reflectuntur, tum à perip heria inter ſemidiametros, in
            <lb/>
          quibus ſunt: tum ab alia huic oppoſita: à reliquis uerò duab{us} minimè. 20 p 8.</head>
          <p>
            <s xml:id="echoid-s10789" xml:space="preserve">SI uerò b punctũ uiſum fuerit extra diametrum d a g, ducatur diameter tranſiens per b:</s>
            <s xml:id="echoid-s10790" xml:space="preserve"> quæ fit
              <lb/>
            t q.</s>
            <s xml:id="echoid-s10791" xml:space="preserve"> Dico, quòd b poteſt reflecti ad uiſum a per arcum interiacentẽ diametros, in quibus ſunt a
              <lb/>
            & b, & ſimiliter per eius oppoſitum, id eſt, per arcum t d, & per arcum g q:</s>
            <s xml:id="echoid-s10792" xml:space="preserve"> & non poterit refle-
              <lb/>
            cti ab aliquo puncto arcus g t uelarcus q d.</s>
            <s xml:id="echoid-s10793" xml:space="preserve"> Verbi gratia:</s>
            <s xml:id="echoid-s10794" xml:space="preserve"> ſumatur punctum in arcu g t, propet, quod
              <lb/>
            ſit k:</s>
            <s xml:id="echoid-s10795" xml:space="preserve"> & du cantur lineæ a k, k b:</s>
            <s xml:id="echoid-s10796" xml:space="preserve"> donec cadat k b ſuper diametrum d g in puncto o.</s>
            <s xml:id="echoid-s10797" xml:space="preserve"> Cum igitur o & a
              <lb/>
            ſint ex eadẽ parte centri circuli, quod eſt e:</s>
            <s xml:id="echoid-s10798" xml:space="preserve"> perpẽdicularis ducta à puncto k ad e, nõ diuidet angulũ
              <lb/>
            o k a.</s>
            <s xml:id="echoid-s10799" xml:space="preserve"> Et ita b non reflectetur ad a à puncto k.</s>
            <s xml:id="echoid-s10800" xml:space="preserve"> Simili-
              <lb/>
              <figure xlink:label="fig-0171-01" xlink:href="fig-0171-01a" number="105">
                <variables xml:id="echoid-variables95" xml:space="preserve">t k m b f d a o e g c h q</variables>
              </figure>
            ter ſumpto alio pũcto, quod ſit f:</s>
            <s xml:id="echoid-s10801" xml:space="preserve"> patebit, quòd per-
              <lb/>
            pendicularis e f non diuidet angulum a fb.</s>
            <s xml:id="echoid-s10802" xml:space="preserve"> Et ita nõ
              <lb/>
            reflectetur b ad a à puncto f.</s>
            <s xml:id="echoid-s10803" xml:space="preserve"> Quòd autem à puncto
              <lb/>
            arcus t d, uel arcus g q poſsit fieri reflexio:</s>
            <s xml:id="echoid-s10804" xml:space="preserve"> palàm per
              <lb/>
            hoc.</s>
            <s xml:id="echoid-s10805" xml:space="preserve"> Sit m punctum arcus t d:</s>
            <s xml:id="echoid-s10806" xml:space="preserve"> & ducantur lineę a m,
              <lb/>
            m b:</s>
            <s xml:id="echoid-s10807" xml:space="preserve"> fiet quadrangulum a m b e.</s>
            <s xml:id="echoid-s10808" xml:space="preserve"> Igitur perpendicu-
              <lb/>
            laris e m diuidet angulum a m b.</s>
            <s xml:id="echoid-s10809" xml:space="preserve"> Simili modo ſit h
              <lb/>
            punctum arcus g q:</s>
            <s xml:id="echoid-s10810" xml:space="preserve"> Linea a h ſecabit diametrum t q
              <lb/>
            in puncto c:</s>
            <s xml:id="echoid-s10811" xml:space="preserve"> & linea h b eundem in puncto b.</s>
            <s xml:id="echoid-s10812" xml:space="preserve"> Et ſunt
              <lb/>
            hæc etiam duo puncta ex diuerſis partibus centri.</s>
            <s xml:id="echoid-s10813" xml:space="preserve">
              <lb/>
            Quare linea e h diuidet illum angulum.</s>
            <s xml:id="echoid-s10814" xml:space="preserve"> Pari modo,
              <lb/>
            ſi fuerit b in ſuperficie ſpeculi:</s>
            <s xml:id="echoid-s10815" xml:space="preserve"> aut extra ſpeculum,
              <lb/>
            dum a ſit intra ſpeculum:</s>
            <s xml:id="echoid-s10816" xml:space="preserve"> idem erit probãdi modus,
              <lb/>
            qui prius.</s>
            <s xml:id="echoid-s10817" xml:space="preserve"> Similiter ſi a fuerit in ſuperficie ſpeculi, b
              <lb/>
            interius, aut exterius.</s>
            <s xml:id="echoid-s10818" xml:space="preserve"> Si uerò a fuerit extra ſpecu-
              <lb/>
            lum, b intra:</s>
            <s xml:id="echoid-s10819" xml:space="preserve"> patebit, quod diximus.</s>
            <s xml:id="echoid-s10820" xml:space="preserve"> Ducantur enim lineæ à puncto a contingentes circulum d t g
              <lb/>
            [per 17 p 1] quæ ſint a h, a z:</s>
            <s xml:id="echoid-s10821" xml:space="preserve"> & ducantur duę diametri
              <lb/>
              <figure xlink:label="fig-0171-02" xlink:href="fig-0171-02a" number="106">
                <variables xml:id="echoid-variables96" xml:space="preserve">a z m d h f b t b e q q g</variables>
              </figure>
            a e g, t e q:</s>
            <s xml:id="echoid-s10822" xml:space="preserve"> & b in diametro t e q:</s>
            <s xml:id="echoid-s10823" xml:space="preserve"> reflectetur b ad a ab
              <lb/>
            aliquo puncto arcus t d:</s>
            <s xml:id="echoid-s10824" xml:space="preserve"> [ut conftat è iam demonſtra
              <lb/>
            tis.</s>
            <s xml:id="echoid-s10825" xml:space="preserve">] Sed palàm, quòd non ab aliquo puncto arcus z
              <lb/>
            d.</s>
            <s xml:id="echoid-s10826" xml:space="preserve"> [ductis enim duabus rectis e z, b z:</s>
            <s xml:id="echoid-s10827" xml:space="preserve"> erit angulus e z
              <lb/>
            a rectus per 18 p 3, & e z b acutus, ut oſtenſum eſt 60
              <lb/>
            n.</s>
            <s xml:id="echoid-s10828" xml:space="preserve"> Quare ob angulorum inæquabilitatem, à puncto
              <lb/>
            z, ad uiſum a nulla fiet reflexio:</s>
            <s xml:id="echoid-s10829" xml:space="preserve"> multò igitur minus à
              <lb/>
            punctis inter z & d intermedijs:</s>
            <s xml:id="echoid-s10830" xml:space="preserve"> quia angulorum ad
              <lb/>
            lineã z a factorũ, unius quidẽ acuti, alterius uerò ob-
              <lb/>
            tuſi per 16 p 1, multò maior futura eſt inęquabilitas.</s>
            <s xml:id="echoid-s10831" xml:space="preserve">]
              <lb/>
            Igitur ab aliquo pũcto arcus t z:</s>
            <s xml:id="echoid-s10832" xml:space="preserve"> & ſimiliter ab aliquo
              <lb/>
            pũcto arcus oppoſiti ipſi t d, ſcilicet arcus g q refle-
              <lb/>
            xio fiet.</s>
            <s xml:id="echoid-s10833" xml:space="preserve"> Sed ab arcu t g, uel d q nõ fiet reflexio ſecun-
              <lb/>
            dũ ſuprà dictũ modũ.</s>
            <s xml:id="echoid-s10834" xml:space="preserve"> Si uerò b fuerit extra hanc dia
              <lb/>
            metrũ, & ſuper aliã, quæ ſimiliter ſit t e q:</s>
            <s xml:id="echoid-s10835" xml:space="preserve"> fiet reflexio
              <lb/>
            ab arcu t d:</s>
            <s xml:id="echoid-s10836" xml:space="preserve"> & à ſola parte eiust z, & ab arcu oppoſito,
              <lb/>
            qui eſt g q:</s>
            <s xml:id="echoid-s10837" xml:space="preserve"> ſed ab arcu t g, uel d q non fiet reflexio.</s>
            <s xml:id="echoid-s10838" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div390" type="section" level="0" n="0">
          <figure number="107">
            <variables xml:id="echoid-variables97" xml:space="preserve">l p m t n b d a c g x s u q</variables>
          </figure>
          <head xml:id="echoid-head370" xml:space="preserve" style="it">67. Si uiſu & uiſibili in diuerſis diametris circuli (qui eſt communis ſectio ſuperficierum, re-
            <lb/>
          flexionis & ſpeculi ſphærici caui) ſitis: linea à uiſu parallela dia-
            <lb/>
          metro uiſibilis, ſecet dicti circuli peripheriam. Imago reflexa à peripheria inter parallelam & uiſibilis diametrum, uidebitur extra ſpeculum: à peripheria inter par allelam & diametrum ui- ſ{us}, ultra uiſum: à peripheria uerò oppoſita, inter uiſum & ſpe- culum. 21 p 8.</head>
          <p>
            <s xml:id="echoid-s10839" xml:space="preserve">VErũ ſi â puncto a ducatur æquidiſtans t e:</s>
            <s xml:id="echoid-s10840" xml:space="preserve"> quæ ſit a p:</s>
            <s xml:id="echoid-s10841" xml:space="preserve"> loca ima
              <lb/>
            ginũ reflexarũ à punctis arcus t p, erunt extra ſpeculũ:</s>
            <s xml:id="echoid-s10842" xml:space="preserve"> loca au
              <lb/>
            tẽ imaginũ arcus p d, ultra centrũ uiſus, quod eſt a:</s>
            <s xml:id="echoid-s10843" xml:space="preserve"> loca au-
              <lb/>
            tem imaginum arcus q g ſunt inter centrum uiſus & ſpeculum.</s>
            <s xml:id="echoid-s10844" xml:space="preserve"> Et
              <lb/>
            quod ſuprà [60.</s>
            <s xml:id="echoid-s10845" xml:space="preserve"> 61 n] dictum eſt de locis imaginum:</s>
            <s xml:id="echoid-s10846" xml:space="preserve"> idem intelligẽ-
              <lb/>
            dum, ducta a m æquidiſtante lineæ t q.</s>
            <s xml:id="echoid-s10847" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div391" type="section" level="0" n="0">
          <head xml:id="echoid-head371" xml:space="preserve" style="it">68. In quolibet puncto diametri circuli (qui eſt com-
            <lb/>
          munis ſectio ſuperficierum, reflexionis & ſpeculi ſphæri-
            <lb/>
          ci caui) quantumlibet continuatæ, poteſt imago uideri.
            <lb/>
          22 p 8.</head>
          <p>
            <s xml:id="echoid-s10848" xml:space="preserve">AMplius:</s>
            <s xml:id="echoid-s10849" xml:space="preserve"> ſumpta diametro circuli in ſphærico ſpeculo cõcau:</s>
            <s xml:id="echoid-s10850" xml:space="preserve"> quodlibet punctũ illius diametri,
              <lb/>
            </s>
          </p>
        </div>
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